The longest increasing subsequence of Brownian separable permutons

Abstract: We establish a scaling limit result for the length $\operatorname{LIS}(\sigma_n)$ of the longest increasing subsequence of a permutation $\sigma_n$ of size $n$ sampled from the Brownian separable permuton $\boldsymbol{\mu}_p$ of parameter $p\in(0,1)$, which is the universal limit of pattern-avoiding permutations. Specifically, we prove that [\frac{\operatorname{LIS}(\sigma_n)}{n^\alpha}\;\underset{n\to\infty}{\overset{\mathrm{a.s.}}{\longrightarrow}}\; X,] where $\alpha=\alpha(p)$ is the unique solution in the interval $(1/2,1)$ to the equation [\frac{1}{4^{\frac{1}{2\alpha}}\sqrt{\pi}}\,\frac{\Gamma\big(\tfrac{1}{2}-\tfrac{1}{2\alpha}\big)}{\Gamma\big(1-\tfrac{1}{2\alpha}\big)}=\frac{p}{p-1},] and $X=X(p)$ is a non-deterministic and a.s. positive and finite random variable, which is a measurable function of the Brownian separable permuton. Notably, the exponent $\alpha(p)$ is an increasing continuous function of $p$ with $\alpha(0<sup>+)=1/2$, $\alpha(1<sup>-)=1$ and $\alpha(1/2)\approx0.815226$, which corresponds to the permuton limit of uniform separable permutations. We prove analogous results for the size of the largest clique of a graph sampled from the Brownian cographon of parameter $p\in(0,1)$.

• The paper establishes that the longest increasing subsequence scales as n^α with a precise exponent derived from gamma function formulations, deepening our theoretical understanding.
• It employs the Rèmy algorithm for tree coupling to break down complex permutation structures into concrete numerical scaling properties.
• The study reveals a duality by drawing parallels to scaling laws in Brownian cographons, where largest cliques and independent sets follow analogous patterns.

An Overview of the Longest Increasing Subsequence of Brownian Separable Permutons

This paper, authored by Adhikari, Borga, Budzinski, Da Silva, and Sénizergues, offers substantial findings in the study of permutation scaling limits, specifically concerning the longest increasing subsequence (LIS) within permutations sampled from the Brownian separable permuton. The results have implications for both permutation theory and the dense graph limit theory through its parallel examination of cographons.

Introduction and Objectives

The central object of examination is the Brownian separable permuton, a probabilistic limit object that emerges as a universal scaling limit for various families of pattern-avoiding permutations. Concurrently, the study considers the Brownian cographon, the analogous limit for certain dense random graphs, particularly focusing on the size of their largest cliques and independent sets.

Key questions addressed include:

• What is the asymptotic behavior of $LIS(\sigma_n)$, where $\sigma_n$ is a random permutation of size $n$ sampled from the Brownian separable permuton?
• How does the size of the largest clique $LCL(G_n)$ of a graph $G_n$ sampled from the Brownian cographon scale?

Main Results

1. Theorem on the LIS of Brownian Separable Permutons: The paper establishes that the LIS of permutations from the Brownian separable permuton scales with the size of the permutation according to a specific exponent $\alpha(p)$. This exponent is a solution in the interval $(1/2,1)$ of a complex equation involving gamma functions and describes how the LIS grows with $n$ as $n^\alpha$ multiplied by a random variable $X(p)$.
2. Parallel Result for Brownian Cographons: Analogous results are demonstrated for the Brownian cographon, showing that both the largest clique and the largest independent set sizes scale similarly with a different exponent but related to the permuton model by $\sigma_n$0 and $\sigma_n$1, showing a duality in behavior due to the parameter $\sigma_n$2.

Methodological Approach

The approach involves using the Rèmy algorithm to couple trees of different sizes, ensuring that the statistical properties for LIS can be traced through these structured decompositions. The authors also develop a detailed coupling strategy enabling a deeper understanding of the convergence properties, operationalizing a unique reductionist method that allows abstract properties of large permutations and graphs to be translated into concrete numerical results.

Implications and Future Directions

Theoretical Implications:

The findings have broad implications for understanding universality classes in combinatorial structures, linking permutation patterns to detailed descriptions in terms of stochastic processes.

Practical Implications:

The theoretical framework developed here potentially impacts algorithmic approaches, where understanding LIS formation in related stochastic structures may improve performance in sorting and routing algorithms.

Future Directions:

The authors suggest that further studies might extend these findings to more generalized permutation models, possibly examining additional metrics such as breadth-first search distances or connectivity thresholds in graphon models, opening up new areas for research in random structure theory.

Conclusions

This work provides a meticulous examination of the asymptotic characteristics of permutons and graphons. By elucidating these scaling limits, it solidifies the position of the Brownian separable permuton and cographon as central objects of interest in understanding complex random structures.